Andrew Haas
We study metrical properties of various subsequences associated to the sequence of rational approximants coming from the continued fraction of an irrational number. Our methods build upon Bosma, Jager and Wiedijk's proof of the Doblin-Lenstra conjecture as well as Jager's subsequent treatment of the sequence of approximation pairs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let F : P^N --> P^N be a dominant rational map. The dynamical degree of F is the quantity d_F = lim (deg F^n)^(1/n). When F is defined over a number field, we define the arithmetic degree of an algebraic point P to be a_F(P) = limsup h(F^n(P))^(1/n) and the canonical height of P to be h_F(P) = limsup h(F^n(P))/n^k d_F^n for an appropriately chosen integer k = k_F. In this article we prove some elementary relations and make some deep conjectures relating d_F, a_F(P), and h_F(P). We prove our conjectures for monomial maps.
We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps…
Let $d\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the rational map $f_t$. We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height.
Given two rational maps $φ$ and $ψ$ on $\PP^1$ of degree at least two, we study a symmetric, nonnegative-real-valued pairing $<φ,ψ>$ which is closely related to the canonical height functions $h_φ$ and $h_ψ$ associated to these maps. Our main results show a strong connection between the value of $<φ,ψ>$ and the canonical heights of points which are small with respect to at least one of the two maps $φ$ and $ψ$. Several necessary and sufficient conditions are given for the vanishing of $<φ,ψ>$. We give an explicit upper bound on the difference between the canonical height $h_ψ$ and the standard…
Let K be a function field, let f be a rational function of degree d at least 2 defined over K, and suppose that f is not isotrivial. In this paper, we show that a point P in P^1(Kbar) has f-canonical height zero if and only if P is preperiodic for f. This answers affirmatively a question of Szpiro and Tucker, and generalizes a recent result of Benedetto from polynomials to rational functions. We actually prove the following stronger result, which is a variant of the Northcott finiteness principle: there exists epsilon > 0 such that the set of points P in P^1(K) with f-canonical height at most …
Canonical singularities on moduli spaces of rational curves via the circle method
本文发展了圆法版本，证明了在维度相对于次数足够大时，光滑超曲面上的有理曲线空间仅含典范奇点。
On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$
研究\(\mathbb{P}^{1}\)上多项式映射的典范高度与朴素高度差，建立与次数\(d\)无关的精确界，以此确定\(\mathbb{Q}\)上高度为0的多项式映射的有理预周期点集并分类其画像。
Fix a rational basepoint b and a rational number c. For the quadratic dynamical system f_c(x) = x^2+c, it has been shown that the number of rational points in the backward orbit of b is bounded independent of the choice of rational parameter c. In this short note we investigate the dependence of the bound on the basepoint b, assuming a strong form of the Mordell Conjecture.
In this paper we investigate the behavior of trajectories of one class of rational $p$-adic dynamical systems in complex $p$-adic field $\C_p$. We studied Siegel disks and attractors of such dynamical systems. We found the basin of the attractor of the system. It is proved that such dynamical systems are not ergodic on a unit sphere with respect to the Haar measure.
We introduce and study algebraic dynamical systems generated by triangular systems of rational functions. We obtain several results about the degree growth and linear independence of iterates as well as about possible lengths of trajectories generated by such dynamical systems over finite fields. Some of these results are generalisations of those known in the polynomial case, some are new even in this case.
Let K be a function field in one variable over an arbitrary field F. Given a rational function f(z) in K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of f all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be non-preperiodic points of canonical height zero. In this paper, we show that for polynomial f, such points exist only if f is isotrivial. In fact, such K-rational points …
We compute a lower bound of the canonical height on quadratic twists of certain elliptic curves. Also we show a simple method for constructing families of quadratic twists with an explicit rational point. % from cubic polynomials. Using the above lower bound, we show that the explicit rational point is primitive as an element of the Mordell--Weil group.
Given a rational function of degree at least two defined over a number field k, we study the cardinality of the set of rational iterated preimages. We prove bounds for the cardinality of this set as the rational function varies in certain families. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for iterated preimages of rational functions and relate this conjecture to other well-known conjectures in arithmetic dynamics.
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa's conjecture on exponential sums, with the log-canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for non-rational singularities, by comparing the log canonical threshold with a local notion of the motivic oscillation index.
Let $K$ denote a number field and a finite set $S$ of places of $K$ and $ϕ:\PP^n\rightarrow\PP^n$ be rational morphism defined over $K$. The main result of this paper proves that there are only finitely many twists of $ϕ$ defined over $K$ which have good reduction at all places outside $S$. This answers a question of Silverman in the affirmative.
Let $f \in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height $\hat{h}_f(x)$ of a non-preperiodic rational point $x$ is bounded below by a uniform multiple of the height of $f$ itself. We provide support for these conjectures by computing the set of preperiodic and small height rational points for a set of degree 2 maps far beyond the range …
Universality of rational canonical form for random matrices over a finite field
研究有限域上随机矩阵有理规范形的分布，证明其渐进行为符合Cohen-Lenstra分布，恢复Fulman的均匀情况。
Rational curves on toroidal compactifications of Picard modular varieties
该研究改进了皮卡模簇环面紧化典范丛丰富性的维数界，构造了含特定有理曲线的无穷多不可公度非紧皮卡模簇，证明相关曲线在任意高维均存在。
We show that if f is a nonzero, noninvertible function on a smooth complex variety X and J_f is the Jacobian ideal of f, then lct(f, J_f^2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if this is not the case, then lct(f, J_f^2)=lct(f). We give two proofs, one relying on arc spaces and one that shows that the minimal exponent of f is at least as large as lct(f, J_f^2). In the case of a polynomial over the algebraic closure of Q, we also prove an analogue of this latter inequality, with the minimal exponent replaced by the motivic oscillation index moi(f)…
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove invariance. The second proof defines the fraction of a rational tangle via integral coloring of the tangle. The coloring method is then used to prove the Kauffman-Harary coloring conjecture for alternating knots and links in the case of rational knots and links (closures of rationa…
We prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.
Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field
该研究确定有限域上稀疏随机矩阵有理典范形式普适性的 sharp 阈值，通过满射矩量法证明阈值与有限域余核、$\mathbb{Z}_p$ 上随机矩阵模型的阈值一致，还构造了对应最小次数不可约多项式的临界稀疏障碍。
We conjecture a lower bound for the minimal canonical height of non-torsion rational points on a natural density 1 subset of the sextic twist family of Mordell curves. We then establish a lower bound that yields a partial result towards this conjecture.
We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times P^1 defined over a number field. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for endomorhisms of P^1\times P^1 defined over the complex numbers.
For one variable rational function $ϕ\in K(z)$ over a field $K$, we can define a discrete dynamical system by regarding $ϕ$ as a self morphism of $\mathbb{P}_{K}^{1}$. Hatjispyros and Vivaldi defined a dynamical zeta function for this dynamical system using multipliers of periodic points, that is, an invariant which indicates the local behavior of dynamical systems. In this paper, we prove the rationality of dynamical zeta functions of this type for a large class of rational functions $ϕ\in K(z)$. The proof here relies on Woods Hole fixed point formula and some basic facts on the trace of a li…
On modular forms of rational weight satisfying the canonical second-order linear modular differential equation
通过将KZ方程转化为超几何微分方程并分析单值性表示，完全分类了使得KZ方程具有由主同余子群Γ(N)的模形式构成的基本解系的有理权k。
We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with 8 integers in an orbit and several minimal cubic rational functions with 10 integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro's conjecture in a dynamical setting and on the structure of the set o…
We construct canonical heights of subvarieties for dynamical system of several morphisms associated with line bundles defined over a number field, and study some of their properties. We also construct invariant currents for such systems over $\mathbb{C}$.
Bigness of Canonical Quadratic Points on Curves of Genus 4
研究四次非双有理曲线的典范二次点，通过模形式和贝蒂映射建立标准条件，证明其在三自映射区域和某些CM族上为大，从而获得非 torsion 理性点和北特里特有限性结果。
Based on continued fractions with subtractions, we identify the set of real numbers with the set of infinite integer sequences with all terms but the first one greater or equal to two. Each such sequence produces in a canonical way a unique strictly decreasing Cauchy sequence of rationals which converges to the corresponding real number. The correspondence is such that the standard order of real numbers is translated to the lexicographic order of sequences.
We formulate a conjecture about the distribution of the canonical height of the lowest non-torsion rational point on a quadratic twist of a given elliptic curve, as the twist varies. This conjecture seems to be very deep and we can only prove partial results in this direction.
We define a new canonical height pairing on the rational points of elliptic curves over global function fields which takes values in the multiplicative group of a completion of the function field. This height serves as an analogue of both the classical Neron-Tate height and analytic p-adic heights. Our main result is that under some general hypotheses this pairing is non-degenerate.
Borel--Serre type bordifications and Canonical Pairs for Loop groups
本文为实环路群对称空间构造Borel–Serre型加边，证明其边界同伦于仿射有理Tits building，并借助Garland约化理论、半稳定性概念研究算术群作用的商空间及其非紧致性。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。